Scattering in twisted waveguides

Abstract

We consider a twisted quantum waveguide i.e. a domain of the form Ω_θ : = r_θω\times R, where ω\subset R^2 is a bounded domain, and r_θ= r_θ(x_3) is a rotation by the angle θ(x_3) depending on the longitudinal variable x_3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian H_θ, self-adjoint in L^2 (Ω_θ), and consider related scattering problems. First, we show that if the derivative of the difference θ_1 - θ_2 decays fast enough as |x_3| goes to infinity, then the wave operators for the operator pair (H_{θ_1}, H_{θ_2}) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. θ' = β- ε, with constant β\in R, and εwhich decays fast enough at infinity together with its first derivative. In this case the unperturbed operator corresponding to εis an analytically fibered Hamiltonian with purely absolutely continuous spectrum. Obtaining Mourre estimates with a suitable conjugate operator, we prove, in particular, that the singular continuous spectrum of H_θ, is empty.